# Mathematical meaning of an inverted yield curve

I am currently working on rates model. I would like to understand, mathematically, what does it mean to have an inverted yield curve? And I am asking myself for a certain model, how can I generate an inverted yield curve? Any content/document is welcome. Thank you

My thoughts:

Using the definition found on Wikipedia:

In finance, an inverted yield curve is a yield curve in which short-term debt instruments (typically bonds) have a greater yield than longer term bonds.

At time $$t$$ we observe yields $$Y(t, T) = \frac{1}{T-t}\int_t^Tf(t, s)ds$$, where $$f(t, s)$$ is the instantaneous forward rate for time $$s$$, observed at time $$t$$. In this setup, an inverted yield curve observed at time $$t$$ is a yield curve $$Y(t, T)$$ that is not increasing in $$T$$. I can provide a proof if desired, but a forward curve $$f(t, T)$$ that is not increasing in $$T$$ is an equivalent condition.

For your second question, my thought would be that a good place to start is with a model that works on forward curves directly, such as HJM-- this model can indeed generate an inverted yield curve by generating a forward curve that is not non-decreasing.

Although the entire shape of the yield curve should perhaps be taken into account, in practice just two maturities are chosen for comparison.

Campbell Harvey in 1985 defined inverted yield curve as: the 3 month yield is larger than the 10 year yield. I.e. the curve is said to be inverted when the difference between these two rates is negative. (See chart)

https://fred.stlouisfed.org/series/T10Y3M

An alternative definition some people use compares the 2 year yield to the 10 year yield.

https://fred.stlouisfed.org/series/T10Y2Y

The simplest interest models produce monotonic yield curves for ZCB's (This is apparent when you look at the formulas for yields of ZCB's in these models). So an inversion can never occur in these models by either definition.